This book is part of the series “Discrete Mathematics and its Applications.” It continues the recent line of books that exploit the connections between the two seemingly disparate subjects of graph theory and matrix theory. While some of these books are more along the lines of graduate-level research monographs (such as An Introduction to the Theory of Graph Spectra by Cvetković, Rowlinson, and Simić), or an undergraduate textbook (Graphs and Matricesby Bapat) , this book works well as a reference textbook for undergraduates. Indeed, it is a distillation of a number of key results involving, specifically, the Laplacian matrix associated with a graph (which is sometimes called the “nodal admittance matrix” by electrical engineers).

After two chapters covering the preliminaries in Matrix Theory and Graph Theory necessary for the sequel, Molitierno presents an Introduction to Laplacian Matrices, with a proof of the Kirchhoff Matrix-Tree Theorem via Cauchy-Binet. He discusses Laplacians of weighted graphs as well as unweighted ones, and bounds on the eigenvalue spectra of certain classes of graphs. In particular, Molitierno focuses on the second smallest eigenvalue of a graph’s Laplacian matrix, called the algebraic connectivity of the graph.

The important work of Grone and Merris is given a decent treatment, as is Fielder’s. In fact, it is Fiedler’s theorem on eigenvectors that leads to a particular type of matrix that dominates the last two chapters of the book, the so-called “bottleneck matrices.” These matrices are used to determine such graph properties as algebraic connectivity. Chapter 6 covers the bottleneck matrices for trees, while some general classes of non-tree graphs are covered in chapter 7.

Molitierno’s book represents a well-written source of background on this growing field. The sources are some of the seminal ones in the field, and the book is accessible to undergraduates.

John T. Saccoman is Professor of Mathematics at Seton Hall University in South Orange, NJ.

Graph Theory PreliminariesIntroduction to GraphsOperations of Graphs and Special Classes of GraphsTreesConnectivity of GraphsDegree Sequences and Maximal GraphsPlanar Graphs and Graphs of Higher Genus

Introduction to Laplacian MatricesMatrix Representations of GraphsThe Matrix Tree TheoremThe Continuous Version of the LaplacianGraph Representations and EnergyLaplacian Matrices and Networks

The Spectra of Laplacian MatricesThe Spectra of Laplacian Matrices Under Certain Graph OperationsUpper Bounds on the Set of Laplacian EigenvaluesThe Distribution of Eigenvalues Less than One and Greater than OneThe Grone-Merris ConjectureMaximal (Threshold) Graphs and Integer SpectraGraphs with Distinct Integer Spectra

The Algebraic ConnectivityIntroduction to the Algebraic Connectivity of GraphsThe Algebraic Connectivity as a Function of Edge WeightThe Algebraic Connectivity with Regard to Distances and DiametersThe Algebraic Connectivity in Terms of Edge Density and the Isoperimetric NumberThe Algebraic Connectivity of Planar GraphsThe Algebraic Connectivity as a Function Genus k where k is greater than 1

The Fiedler Vector and Bottleneck Matrices for TreesThe Characteristic Valuation of VerticesBottleneck Matrices for TreesExcursion: Nonisomorphic Branches in Type I TreesPerturbation Results Applied to Extremizing the Algebraic Connectivity of TreesApplication: Joining Two Trees by an Edge of Infinite WeightThe Characteristic Elements of a TreeThe Spectral Radius of Submatrices of Laplacian Matrices for Trees

Bottleneck Matrices for GraphsConstructing Bottleneck Matrices for GraphsPerron Components of GraphsMinimizing the Algebraic Connectivity of Graphs with Fixed GirthMaximizing the Algebraic Connectivity of Unicyclic Graphs with Fixed GirthApplication: The Algebraic Connectivity and the Number of Cut VerticesThe Spectral Radius of Submatrices of Laplacian Matrices for Graphs

The Group Inverse of the Laplacian MatrixConstructing the Group Inverse for a Laplacian Matrix of a Weighted TreeThe Zenger Function as a Lower Bound on the Algebraic ConnectivityThe Case of the Zenger Equalling the Algebraic Connectivity in TreesApplication: The Second Derivative of the Algebraic Connectivity as a Function of Edge Weight